What Drexel28 is asking is: What does the dot mean in ? It looks as though it is supposed to be an inner product. But a Banach space doesn't have an inner product (unless it is a Hilbert space), so it's hard to know what this problem is supposed to be about.

Xn and Yn are in Banach space. So in a normed linear space Xn converges to X and Yn converges to Y then prove that XnYn (multiply) converges to XY.

That is the full problem.

There is no operation of multiplication defined in a Banach space, so this question makes no sense.

In order for a product to be defined, there must be some additional structure in the space. It has to be either a Hilbert space (with an inner product), or a Banach algebra (with a multiplicative structure).

Also, as HallsofIvy has pointed out, a Banach space is not the same as a normed linear space.

A "Banach space" is a vector space having a norm such that "Cauchy sequences" converge.

While "scalar multiplication" and "vector addition" are defined in any vector space, multiplication of vectors is NOT generally defined. There is no "standard multiply"! It is impossible to answer your question with knowing how you are defining "XnYn".